Question

How do you find the area bounded by the cardioid `r=1+cos(theta)`?

Answer 1

`A=((cosx+4)sinx+3x)/4`

Explanation:

Given: `r=1+cos(theta)`
Required: Area of cardioid?
Solution Strategy: Polar Coordinate Area Integral
`A= int_(theta_1)^(theta_2) 1/2r^2d(theta)` substitute for `r`
`A= 1/2int_(theta_1)^(theta_2) ( 1+cos(theta))^2d(theta)`
`=1/2[int (1+2costheta+cos^2theta) d(theta)]`
`= 1/2[theta + 2sintheta+ int cos^2theta d(theta)]`
`I_3=color(brown)(int cos^2theta d(theta))` apply reduction formula
`I_3=(n-1)/n int cos^(n-2)thetad(theta) +(cos^(n-1)thetasintheta)/n`
`I_3=color(brown)(1/2 theta +(costhetasintheta)/2)`
Putting it all together:

`A=((cosx+4)sinx+3x)/4`